Gibbs Vector Research Articles

Attitude determination for multirotor aerial vehicles using a prescribed‐time super‐twisting algorithm

AbstractThis paper is concerned with the prescribed‐time robust attitude determination (AD) of multirotor aerial vehicles (MAVs) using vector measurements from the local magnetic field and local gravity. To address this problem, we first introduce a novel modified super‐twisting algorithm endowed with the prescribed‐time convergence property. The state of the proposed algorithm is governed by an unbounded time‐varying gain up to the prescribed settling time (PST) and by a function after that. Therefore, after the PST, the new algorithm coincides with the conventional super‐twisting, thus showing robust stability at the origin. This prescribed‐time super‐twisting algorithm (PTSTA) is then applied to the formulation of a three‐stage gyro‐free attitude determination method for MAVs. In the first stage, the classical QUEST algorithm is used to compute a Wahba‐optimal attitude estimate from the vector measurements. In the second stage, the PTSTA is employed in the formulation of a robust state estimator that provides estimates of the attitude Gibbs vector and its rate. Finally, in the third stage, these state estimates as well as the attitude kinematic equation are immediately used to compute the MAV angular velocity. The proposed robust prescribed‐time gyro‐free AD method is evaluated numerically, showing invariance with respect to disturbance and model uncertainty.

Vector methods for determination of rigid body orientation

Vector methods for determining the orientation of a rigid body using information about vectors in the reference and body-related coordinate systems are considered. The analysis is based on the least squares method in vector-matrix form. This approach allows us to consider the application of different methods of solving the problem from a single standpoint: matrix (using the directional cosine matrix), geometric (using the Gibbs vector), quaternion (using the quaternion of rotation). Numerical evaluation of the accuracy of eight orientation determination algorithms obtained on the basis of these methods was performed.

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

Extended WTLS iterative algorithm of 3D similarity transformation based on Gibbs vector

Optimal Factorization of the State-Dependent Riccati Equation Technique in a Satellite Attitude and Orbit Control System

The satellite attitude and orbit control system (AOCS) can be designed with success by linear control theory if the satellite has slow angular motions and small attitude maneuver. However, for large and fast maneuvers, the linearized models are not able to represent all the perturbations due to the effects of the nonlinear terms present in the dynamics and in the actuators (e.g., saturation). Therefore, in such cases, it is expected that nonlinear control techniques yield better performance than the linear control techniques. One candidate technique for the design of AOCS control law under a large maneuver is the State-Dependent Riccati Equation (SDRE). SDRE entails factorization (that is, parameterization) of the nonlinear dynamics into the state vector and the product of a matrix-valued function that depends on the state itself. In doing so, SDRE brings the nonlinear system to a (nonunique) linear structure having state-dependent coefficient (SDC) matrices and then it minimizes a nonlinear performance index having a quadratic-like structure. The nonuniqueness of the SDC matrices creates extra degrees of freedom, which can be used to enhance controller performance, however, it poses challenges since not all SDC matrices fulfill the SDRE requirements. Moreover, regarding the satellite's kinematics, there is a plethora of options, e.g., Euler angles, Gibbs vector, modified Rodrigues parameters (MRPs), quaternions, etc. Once again, some kinematics formulation of the AOCS do not fulfill the SDRE requirements. In this paper, we evaluate the factorization options (SDC matrices) for the AOCS exploring the requirements of the SDRE technique. Considering a Brazilian National Institute for Space Research (INPE) typical mission, in which the AOCS must stabilize a satellite in three-axis, the application of the SDRE technique equipped with the optimal SDC matrices can yield gains in the missions. The initial results show that MRPs for kinematics provides an optimal SDC matrix.

Model of the influence of Internet finance on monetary policy based on gibbs sampling and vector autoregression

The various parts of the traditional financial supervision and management system can no longer meet the current needs, and further improvement is urgently needed. In this paper, the low-frequency data is regarded as the missing of the high-frequency data, and the mixed frequency VAR model is adopted. In order to overcome the problems caused by too many parameters of the VAR model, this paper adopts the Bayesian estimation method based on the Minnesota prior to obtain the posterior distribution of each parameter of the VAR model. Moreover, this paper uses methods based on Kalman filtering and Kalman smoothing to obtain the posterior distribution of latent state variables. Then, according to the posterior distribution of the VAR model parameters and the posterior distribution of the latent state variables, this paper uses the Gibbs sampling method to obtain the mixed Bayes vector autoregressive model and the estimation of the state variables. Finally, this article studies the influence of Internet finance on monetary policy with examples. The research results show that the method proposed in this article has a certain effect.

On the derivatives of curvature of framed space curve and their time-updating scheme

This paper considers the curvature of framed space curves, their higher-order derivatives, variations, and co-rotational derivatives. We realize that parametrizing rotation tensor using the Gibbs vector is effective in deriving a closed-form formula that gives the general n-th order derivative of the curvature tensor as the summation of functions of the parametrizing quantity and its own derivatives. We use these results for formulating a linearized updating algorithm for curvature and its derivatives when the configuration of the curve acquires a small increment.

Analytical and Iterative Solutions to GNSS Attitude Determination Problem in Measurement Domain

Attitude determination using double-differenced GNSS carrier phase measurements is studied. A realistic stochastic model is employed to take the correlations among the double-differenced measurements into full consideration. Two important issues concerning iteratively solving the nonlinear least-squares attitude determination problem are treated, namely, the initial guess and the iteration scheme. An analytical and sub-optimal solution is employed to provide the initial guess. In this solution, the orthogonal and determinant constraints among the elements of the direction cosine matrix (DCM) of the attitude are firstly ignored, and hence a relaxed 3×3 matrix is estimated using the linear weighted least-squares method. Then a mathematically feasible DCM, i.e., orthogonal and with +1 determinant, is extracted from the relaxed matrix estimate, optimally in the sense of minimum Frobenius norm. This analytical initial guess estimation method can be used for all feasible cases, including some generated ones, e.g., the case with only 3 antennas and only 3 satellites, subject possibly to some necessary, yet minor modifications. In each iteration, an error attitude, whose DCM is parameterized using the Gibbs vector, is introduced to relate the previously estimated and the true DCM. By linearizing the measurement model at the zero Gibbs vector, the least-squares estimate of the Gibbs vector is obtained and then used to correct the previously estimated DCM. By repeating this process, the truly least-squares estimate of the attitude can be achieved progressively. These are in fact Gauss-Newton iterations. For the final estimate, the variance covariance matrix (VCM) of the attitude estimation error can be retained to evaluate or predict the estimation accuracy. The extraction of the widely used roll-pitch-yaw angles and the VCM of their additive estimation errors from the final solution is also presented. Numerical experiments are conducted to check the performance of the developed theory. For the case with 3 2-meter long and orthogonally mounted baselines, 5 visible satellites, and 5-millimeter standard deviations of the carrier phase measurements, the root mean squared errors (RMSE) of the roll-pitch-yaw angles in the analytical solution are well below 0.5 degrees, and the estimates converge after only one iteration, with all three RMSEs below 0.2 degrees.

GNSS attitude determination method through vectorisation approach

Determining the attitude using GNSS carrier signals is studied. It features an analytical approach to get an estimate as initial guess for iterative algorithms, in three steps. First, baseline vectors are estimated by least-squares method. Second, the constraint of the direction cosine matrix (DCM) is ignored and the least-squares estimates of its 9 elements are solved. Third, a mathematically feasible DCM estimate is extracted from the above estimated free matrix. An error attitude, formulated using the Gibbs vector, is introduced to relate the previously estimated and the true attitude, and the measurement model becomes a nonlinear function of the Gibbs vector. The Gauss-Newton iteration is employed to solve the least-squares problem with this measurement model. The estimate of the roll-pitch-yaw angles and the variance covariance matrix of their estimation errors are extracted from the final solution. Numerical experiments are conducted. With 3 orthogonally mounted 3-meter baselines, 4 visible satellites, and 5-millimeter standard-deviation of the carrier measurements, the accuracy of the analytical solution can be less than 1° in the root mean squared error (RMSE) sense. The convergence of the iteration is rather fast, the RMSE converges after only one iteration, with the converged RMSE less than 0.1°.

Representation of the rotation parameter estimation errors in the Helmert transformation model

In the Helmert transformation model, the rotation is more difficult to be treated in terms of representation, estimation, and error analysis. First, two classes of representations of the rotation, i.e. the redundant class including the direction cosine matrix and the unit quaternion, and the minimum class including the rotation vector, the Gibbs vector, the modified Rodrigues parameters, and the Euler angles, are reviewed. It is concluded that in general the redundant class should be preferred as they are transcendental-function-free, singularity-free, and discontinuity-free. Second, two classes of estimation errors, i.e. the additive and the multiplicative errors, are defined and compared in detail. While the multiplicative errors are more convenient, the relationship among different representations and the relationship with their additive counterparts are also explored from first principle. It can be seen as a review paper; however, the content concerning the relationship between the additive and the multiplicative errors is believed new.

Optimal linear attitude estimator and its recursive algorithm via geometric analysis

A new optimal linear attitude estimator is proposed for single-point attitude estimation using geometric approach, and a recursive optimal linear attitude estimator is developed through filtering noisy measurements. Dot and cross products are taken in order to eliminate the unknown parameters of relationships between measurements and Gibbs vector. The optimality criterion, which does not coincide with Wahba’s constrained criterion, yields linear attitude estimate. A prior rotation is adopted to avoid singularity which occurs when the principal angle is close to π. The recursive algorithm is achieved for the purpose of improving attitude accuracy using all past measurements. For long-term space missions, memory fading concept is introduced into recursive optimal linear attitude estimator. The optimal relative weighting is obtained through minimizing error propagation, and an efficient modification is proposed to significantly reduce the sudden increase of attitude error of recursive optimal linear attitude estimator in special cases. Numerical simulations show that the estimate of optimal linear attitude estimator is almost identical with that of the famous QUaternion ESTimator, and the accuracy provided by recursive optimal linear attitude estimator is over an order magnitude higher than that of optimal linear attitude estimator or QUaternion ESTimator in most cases.

A Non-Iterative Algorithm to Determine Camera Position and Orientation

Determination of Camera position and orientation is a basic research subject in photogrammetry and computer vision. This paper introduces a new explicit approach to determine the camera-point distances based on the computer algebraic software of Matlab, then derives the basic mathematic model of pose estimation, and puts forward the analytical calculus of rotation matrix based on the Gibbs vector or Rodrigues matrix, finally presents explicit formulae of camera orientation and position. A simulative case is studied, and the result shows the presented algorithm is correct and efficient for arbitrary pose angles image.

A Non-Iterative Algorithm to Determine Camera Position and Orientation

Determination of Camera position and orientation is a basic research subject in photogrammetry and computer vision. This paper introduces a new explicit approach to determine the camera-point distances based on the computer algebraic software of Matlab, then derives the basic mathematic model of pose estimation, and puts forward the analytical calculus of rotation matrix based on the Gibbs vector or Rodrigues matrix, finally presents explicit formulae of camera orientation and position. A simulative case is studied, and the result shows the presented algorithm is correct and efficient for arbitrary pose angles image.

Pynchon, Cohen, and the Crisis of Victorian Mathematics

Reviewed by: Pynchon, Cohen, and the Crisis of Victorian Mathematics Thomas Dechand Daniel J. Cohen, Equations from God: Pure Mathematics and Victorian Faith. Baltimore: The Johns Hopkins UP, 2007. 242 pages. "They tell me you're kind of famous?" "Women at Göttingen form a somewhat beleaguered subset." She looked around. "And what is it you do here again?" "Drink beer, work on my sleep allowance, the usual." "I took you for a mathematician." "Well . . . maybe not your kind. . . ." "Yes? Come, don't be too clever." "All right, then." He squared his shoulders, brushed imaginary beer foam off his almost-matured mustache, and, expecting her to disappear just as quick as beer-foam, winced in apology. "I'm a sort of, hm . . . Vectorist?" Despite the shadow of an intent to flinch, she surprised him instead with a smile which, for all its resemblance to the smiles one gives the afflicted, was still able to turn Kit's extremities to stone. That is, it was some smile. "They teach vectors in America? I'm amazed." "Nothing like what they offer here." "Isn't England where you ought to be now?" as to a naughty child one expected to become, in a short while, naughtier. "Nothing but Quaternions over there." "Oh dear, not the Quarternion Wars again. That is so all rather fading into history now, not to mention folklore. . . . Why should any of you keep at it this way?" "They believe—the Quaternionists do—that Hamilton didn't so much [End Page 1180] figure the system out as receive it from somewhere beyond? Sort of like Mormons only different?" She couldn't tell how serious he was being, but after a decent interval she stepped closer. "Excuse me? It's a vectorial system, Mr. Traverse, it's something for engineers, to help the poor prats visualize what they obviously can't grasp as real maths."1 In his 2006 novel Against the Day, Thomas Pynchon exhumes a late nineteenth-century debate over the relative values of vector analysis (developed by Willard Gibbs and Oliver Heaviside) and quaternions ("discovered" by William Rowan Hamilton, and most fruitfully extended as a tool for physicists by Peter Tait) as competing mathematical notations used in mathematical physics.2 The dialogue above features two fictional protagonists: Kit Traverse, an American vectorist who trained at Yale under Gibbs, and Yashmeen Halfcourt, who is at Göttingen to work on Riemann's zeta-function. Associative but not commutative, a quaternion can be written in the form q = a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, and k are vectors of magnitude √–1 along the x, y, and z axes. While relatively easy to work with, quaternions, unlike vectors, have proven notoriously difficult to think about. The non-commutativity of quaternions initially puzzled many mathematicians who found themselves unable to detect any physical analogies whatsoever. John Graves, described by one historian as "perhaps the mathematician best prepared for accepting" quaternions, wrote to Hamilton that there was "something in the system that gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and endow them with supernatural properties."3 As Yashmeen's description of the debate as the 'Quaternion Wars' might suggest, the rhetoric used by both sides was particularly charged. Gibbs, in an 1888 letter to Thomas Craig, predicted that a "Kampf ums Dasein" was about to commence between the two sides. In one of the opening salvos of that struggle, Tait classed Gibbs as "one of the retarders of Quaternion progress" and labeled Gibbs's vector analysis "a sort of hermaphrodite monster." Such language is not commensurate with a debate merely over a choice of notation. As Gibbs himself recognized, something more important was thought to be at stake: "[Tait's] criticism relates particularly to notations, but I believe that there is a deeper question of notions underlying that of notations."4 Pynchon's novel provides a clue as to the meaning of the deeper question when Kit Traverse notes that Quaternionists believe "Hamilton didn't so much figure the system out as receive it...

Optimal Linear Attitude Estimator

An optimal linear attitude estimator is presented for the case of a single-point real-time estimation of spacecraft attitude using the minimum-element attitude parameterization: Rodrigues (or Gibbs) vector g. The optimality criterion, which does not coincide with Wahba's constrained criterion, is rigorously quadratic and unconstrained. The singularity, which occurs when the principal angle is π, can easily be avoided by one rotation. The attitude accuracy tests show that the proposed method provides a precision comparable with those fully complying with the Wahba optimality definition. Finally, computational speed tests demonstrate that the proposed method belongs to the class of the fastest optimal attitude estimation algorithms.

Two types of structural domains in Pb3(PO4)2

The conditions of the formation of suborientation states in multidomain and single-domain Pb3(PO4)2 crystals are analyzed. It is shown that suborientation states belong to sets of two structurally different types of domains differing in the angle sign and the orientation of the axis of rotation with respect to the coordinate system of the paraelastic phase. These structural differences are proposed to be described by the Gibbs vector. It is concluded that this macroscopic parameter corresponds to cooperative displacement of some groups of atoms with respect to other groups, with the crystal matrix being at rest. It is found that the modulus of the Gibbs vector is proportional to the spontaneous-strain components and depends linearly on the crystallographic parameter c in the ferroelastic phase.

Nonlinear Predictive Control of Spacecraft

A new approach for the control of a spacecraft with large angle maneuvers is presented. This new approach is based on a nonlinear predictive control scheme which determines the required torque input so that the predicted responses match the desired trajectories. This is accomplished by minimizing the norm-squared local errors between the predicted and desired quantities. Formulations are presented which use either attitude and rate tracking or attitude-tracking solely. The robustness of the new controller with respect to large system uncertainties is also demonstrated. Finally, simulations results are shown which use the new control strategy to stabilize the motion of the Microwave Anisotropy Probe spacecraft. Introduction The control of spacecraft for large angle slewing maneuvers poses a difficult problem. Some of these difficulties include: the governing equations have highly nonlinear characteristics, control rate and saturation constraints and limits, and incomplete state knowledge due to sensor failure or omission. The control of spacecraft with large angle slews can be accomplished by either open-loop or closed-loop schemes. Open-loop schemes usually require a predetermined pointing maneuver and are typically determined using optimal control techniques, which involve the solution of a two-point boundary value problem (e.g., see the time optimal maneuver problem [1]). Also, open-loop schemes are sensitive to spacecraft parameter uncertainties and unexpected disturbances [2]. Closed-loop systems can account for parameter uncertainties and disturbances, and thus provide a more robust design methodology. In recent years, much effort has been devoted to the closed-loop design of spacecraft with large angle slews. Wie and Barber [3] derive a number of simple control schemes using quaternion and angular velocity (rate) feedback. Asymptotic stability is also shown by Copyright c 1997 by John L. Crassidis. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. using a Lyapunov function analysis for all cases. Tsiotras [4] expands upon these formulations by deriving simple control laws based on both a Gibbs vector parameterization and a modified Rodrigues parameterization each with rate feedback (for a complete survey of attitude parameterizations, see [5]). Lyapunov functions are shown for all the controllers developed in [4] as well. Other full state feedback techniques have been developed which are based on sliding mode (variable structure) control, which uses a feedback linearizing technique and an additional term aimed at dealing with model uncertainty [6]. This type of control has been successfully applied for large angle maneuvers using a Gibbs vector parameterization [7], a quaternion parameterization [8], and a modified Rodrigues parameterization [9]. Another robust control scheme using a nonlinear H∞ control methodology has been developed by Kang [10]. This scheme involves the solution of Hamilton-Jacobi-Isaacs inequalities, which essentially determines feedback gains for the full state feedback control problem so that the spacecraft is stabilized in the presence of uncertainties and disturbances. Another class of controllers involves adaptive techniques, which update the model during operation based on measured performances (e.g., see [6]). An adaptive scheme which estimates for external torques by means of tracking a Lyapunov function has been developed by Schaub et. al. [11]. This method has been shown to be very robust in the presence of spacecraft modeling errors. The aforementioned techniques all utilize full state knowledge (i.e., attitude and rate feedback). The problem of controlling a spacecraft without full state feedback becomes increasingly complex. The basic approaches used to solve this problem can be divided into methods which estimate for the unmeasured states using a filter algorithm, or methods which develop control laws directly from output feedback. Filtering methods, such as the extended Kalman filter, have been successfully applied on numerous spacecraft systems without the use of rate-integrating gyro measurements (e.g., see [12]-[14]). An advantage of these methods is that the attitude may be estimated by using only one set of vector attitude measurements (such as magnetometer measurements). However, these methods are usually

Minimal parameter solution of the orthogonal matrix differential equation

The straightforward solution of the first-order differential equation satisfied by all nth-order orthogonal matrices requires n/sup 2/ integrations to obtain the matrix elements. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation, and expressing the orthogonal matrix in terms of these parameters are considered in the present work. Several possibilities which are based on attitude determination in three dimensions are examined. It is concluded that not all 3-D methods have useful extensions to other dimensions, and that the 3-D Gibbs vector (or Cayley parameters) provide the most useful extension. An algorithm is developed using the resulting parameters, which are termed extended Rodrigues parameters, and numerical results are presented of the application of the algorithm to a fourth-order matrix.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Vector cross products on manifolds

possesses an almost complex structure. The properties of these manifolds have been investigated by Calabi [6], Gray [11], and Yano and Sumitomo [22]. The almost complex structure, which is a generalization of the one on S6, is defined in terms of a vector cross product on R7, which is a generalization of the ordinary Gibbs vector cross product on R3. In this paper we give a general definition of vector cross product (?2), and then study vector cross products on manifolds. Vector cross products are interesting for three reasons: first, they are themselves natural generalizations of the notion of almost complex structure; secondly, a vector cross product on a manifold M gives rise to unusual almost complex structures on certain submanifolds of M; thirdly, vector cross products provide one approach to the study of Riemannian manifolds with holonomy group G2 or Spin (7). Vector cross ptoducts on vector spaces have been studied from an algebraic standpoint in Brown and Gray [5] and from a topological standpoint in Eckmann [9] and Whitehead [20]. We consider the topological existence of vector cross products on manifolds in ?2. Then in ?3 we give conditions for the existence of vector cross products on vector bundles over CW-complexes in terms of characteristic classes. As an extra feature of our investigations we determine the behavior of the triality automorphism of Spin (8) on Spin characteristic classes. In ?4 we discuss vector cross products from a differential geometric point of view and in ?5 we develop some important relations between vector cross products and curvatures. The rest of the paper (?6 and ?7) is devoted to almost complex structures on manifolds. We show in ?6 that every orientable 6-dimensional submanifold of R8 possesses an almost complex structure. Such an almost complex structure is defined by means of a 3-fold vector cross product on R8. We also show that all of Calabi's manifolds can be obtained by this method. However, the almost complex structures constructed here are more general in several ways. In the first place, we consider 6-dimensional orientable submanifolds M of arbitrary 8-dimensional pseudo-Riemannian manifolds M possessing 3-fold vector cross products, e.g., parallelizable 8-dimensional manifolds. Secondly, the metric of M we use may be indefinite provided it has signature (4, 4). (Here we

Development of Vector Analysis from Quaternions

About the middle of last century, there were invented several forms of multiple algebra, some of which are extensively used today in mathematical physics. The two types discussed here are quaternions, a quadruple algebra, and vector analysis, a triple algebra. The notations and forms of analysis used in the two systems are contrasted, using for each their representation of scalar and vector products and of Gauss's, Stokes's and Green's theorems. Maxwell made use of the vector part of quaternions in his Treatise on Electricity, and it was largely due to this that Gibbs came to invent his form of vector analysis. By 1890, about fifty years after the invention of quaternions, it became apparent that Gibbs's vector analysis was being used in place of quaternions, and this gave rise to a heated controversy in the pages of Nature. In the seventy years since this controversy, it is apparent that Gibbs's work and point of view have been completely vindicated.

Errata [The orbit of Swift's comet 1880 V, determined by Gibbs's vector method

Errata [The orbit of Swift's comet 1880 V, determined by Gibbs's vector method